Question

A simplified model of a bicycle of mass $M$ has two tires that each comes into contact with the ground at a point. The wheel base of this bicycle is $W$, and the centre of mass $C$ of the bicycle is located midway between the tires and a height $h$ above the ground. The bicycle is moving to the right, but slowing down at a constant acceleration $a$. Air resistance may be ignored. Assuming that the coefficient of sliding friction between each tyre and the ground is $\mu$ and that both tyres are skidding (sliding without rotating). Express your answer in terms of $w, h, M$ and $g$.

What is the maximum value of $\mu$ so that both tires remain in contact with the ground :

• A. $W / 2 h$
• B. $h / 2 W$
• C. $2 h / W$
• D. $W/h$

Answer 1

Answer: (A)

If $N_{1}$ and $N_{2}$ are normal reaction on rear & front tyres, we have
$N_{1}+N_{2}=M g\,\,\,...(1)$
$\mu\left(N_{1}+N_{2}\right)=M a\,\,\,...(2)$
As net torque about $C=0$
$\Rightarrow \mu\left(N_{1}+N_{2}\right) h+\frac{N_{1} W}{2}=\frac{N_{2} W}{2}\,\,\, ...(3)$
$\Rightarrow N_{2}>N_{1}$
Solving equation we get $ N_{1}=\frac{N}{2}\left\{g-\frac{2 \mu g h}{W}\right\}\,\,\, ...(4)$
For maintaning contact $ N_{1} >0$
$ \Rightarrow \mu=\frac{W}{2 h}$